%% boi-5.tex (short version)
%% Chapter 5 of the Book of Involutions

\chapter{Algebras of Degree Three}
\label{5.chap.alg.three}

The main topic of this chapter is central simple algebras of
degree~$3$ with involutions of the second kind and their \'etale
(commutative) subalgebras. To every involution of the second kind on a
central simple algebra of degree~$3$, we attach a $3$-fold Pfister
form, and we show that this quadratic form classifies involutions up
to conjugation. Involutions whose associated Pfister form is
hyperbolic form a distinguished conjugacy class; we show that such an
involution is present on every central simple algebra of degree~$3$
with involution of the second kind, and we characterize distinguished
involutions in terms of \'etale subalgebras of symmetric elements.

We start with Galois descent, followed by a general discussion of
\'etale and Galois algebras. These are tools which will also be used
in subsequent chapters.

\section{{\setbox0\hbox{\'Etale}\ht0=0pt\box0} and Galois Algebras}
%% smashing to avoid overfull vbox messages
\label{etalgal.sec}

Throughout this section, let $F$ be an arbitrary base field, let $\Fs$
be a separable closure of~$F$ and let $\Gamma$\index{B}{Gam@{$\Gamma$
    (absolute Galois group)}} be the \emph{absolute Galois group} of $F$:
\index{A}{absolute Galois group}
\[
\Gamma=\Gal(\Fs/F).
\]
The central theme of this section is a correspondence between \'etale
$F$-algebras and $\Gamma$-sets, which is set up in the first
subsection. This correspondence is then restricted to Galois algebras
and torsors.  The final subsection demonstrates the special features
of \'etale algebras of dimension~$3$.

%% end of boi-5.tex

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "boi"
%%% End: 
